A sled is initially given a shove up a frictionless 27.0° incline. It reaches a maximum vertical height 1.45 m higher than where it started. What was its initial speed?
To determine the initial speed of the sled, we can use the principle of conservation of mechanical energy. This principle states that the total mechanical energy of an object remains constant as long as only conservative forces (like gravity) are acting upon it.
In this case, we can consider the sled as it starts at the initial position with an initial speed (which we want to find), and then reaches a maximum height.
The equation for the conservation of mechanical energy is:
Initial Kinetic Energy + Initial Potential Energy = Final Kinetic Energy + Final Potential Energy
The initial kinetic energy of the sled is given by:
Kinetic Energy = (1/2) * mass * (initial velocity)^2
The initial potential energy of the sled is given by:
Potential Energy = mass * g * (initial height)
The final kinetic energy of the sled is given by:
Final Kinetic Energy = (1/2) * mass * (final velocity)^2
The final potential energy of the sled is given by:
Final Potential Energy = mass * g * (final height)
Since the sled reaches a maximum vertical height of 1.45 m higher than its initial height, we have:
Final Height = Initial Height + 1.45 m
Since the sled starts from rest (no initial velocity), the final velocity at the maximum height is 0 m/s. Therefore, the final kinetic energy is 0.
Using these equations, we can set up the conservation of mechanical energy equation:
(1/2) * mass * (initial velocity)^2 + mass * g * (initial height) = 0 + mass * g * (final height)
With some rearrangement, we can solve for the initial velocity:
(initial velocity)^2 = 2 * g * (final height - initial height)
initial velocity = √(2 * g * (final height - initial height))
Now we can substitute the known values into the equation:
Angle of the incline (θ) = 27.0°
(initial height) = 0 (since the sled starts at the ground)
(final height) = 1.45 m
(gravitational acceleration) = 9.8 m/s^2
Using trigonometry, we can find the vertical height as:
(final height) = (distance) * sin(θ)
Therefore:
(distance) = (final height) / sin(θ)
Now we can calculate the initial velocity:
initial velocity = √(2 * g * (final height - initial height))
= √(2 * 9.8 * ((final height) - 0))
= √(19.6 * (final height))
After substituting the values:
θ = 27.0°,
(final height) = 1.45 m,
(g) = 9.8 m/s^2,
We can calculate the initial velocity.